3.67/1.83	YES
3.67/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.67/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.67/1.84	
3.67/1.84	
3.67/1.84	Termination w.r.t. Q of the given QTRS could be proven:
3.67/1.84	
3.67/1.84	(0) QTRS
3.67/1.84	(1) QTRSToCSRProof [SOUND, 0 ms]
3.67/1.84	(2) CSR
3.67/1.84	(3) CSRRRRProof [EQUIVALENT, 59 ms]
3.67/1.84	(4) CSR
3.67/1.84	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.67/1.84	(6) CSR
3.67/1.84	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.67/1.84	(8) YES
3.67/1.84	
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(0)
3.67/1.84	Obligation:
3.67/1.84	Q restricted rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
3.67/1.84	   active(__(X, nil)) -> mark(X)
3.67/1.84	   active(__(nil, X)) -> mark(X)
3.67/1.84	   active(U11(tt)) -> mark(U12(tt))
3.67/1.84	   active(U12(tt)) -> mark(tt)
3.67/1.84	   active(isNePal(__(I, __(P, I)))) -> mark(U11(tt))
3.67/1.84	   active(__(X1, X2)) -> __(active(X1), X2)
3.67/1.84	   active(__(X1, X2)) -> __(X1, active(X2))
3.67/1.84	   active(U11(X)) -> U11(active(X))
3.67/1.84	   active(U12(X)) -> U12(active(X))
3.67/1.84	   active(isNePal(X)) -> isNePal(active(X))
3.67/1.84	   __(mark(X1), X2) -> mark(__(X1, X2))
3.67/1.84	   __(X1, mark(X2)) -> mark(__(X1, X2))
3.67/1.84	   U11(mark(X)) -> mark(U11(X))
3.67/1.84	   U12(mark(X)) -> mark(U12(X))
3.67/1.84	   isNePal(mark(X)) -> mark(isNePal(X))
3.67/1.84	   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
3.67/1.84	   proper(nil) -> ok(nil)
3.67/1.84	   proper(U11(X)) -> U11(proper(X))
3.67/1.84	   proper(tt) -> ok(tt)
3.67/1.84	   proper(U12(X)) -> U12(proper(X))
3.67/1.84	   proper(isNePal(X)) -> isNePal(proper(X))
3.67/1.84	   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
3.67/1.84	   U11(ok(X)) -> ok(U11(X))
3.67/1.84	   U12(ok(X)) -> ok(U12(X))
3.67/1.84	   isNePal(ok(X)) -> ok(isNePal(X))
3.67/1.84	   top(mark(X)) -> top(proper(X))
3.67/1.84	   top(ok(X)) -> top(active(X))
3.67/1.84	
3.67/1.84	The set Q consists of the following terms:
3.67/1.84	
3.67/1.84	   active(__(x0, x1))
3.67/1.84	   active(U11(x0))
3.67/1.84	   active(U12(x0))
3.67/1.84	   active(isNePal(x0))
3.67/1.84	   __(mark(x0), x1)
3.67/1.84	   __(x0, mark(x1))
3.67/1.84	   U11(mark(x0))
3.67/1.84	   U12(mark(x0))
3.67/1.84	   isNePal(mark(x0))
3.67/1.84	   proper(__(x0, x1))
3.67/1.84	   proper(nil)
3.67/1.84	   proper(U11(x0))
3.67/1.84	   proper(tt)
3.67/1.84	   proper(U12(x0))
3.67/1.84	   proper(isNePal(x0))
3.67/1.84	   __(ok(x0), ok(x1))
3.67/1.84	   U11(ok(x0))
3.67/1.84	   U12(ok(x0))
3.67/1.84	   isNePal(ok(x0))
3.67/1.84	   top(mark(x0))
3.67/1.84	   top(ok(x0))
3.67/1.84	
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(1) QTRSToCSRProof (SOUND)
3.67/1.84	The following Q TRS is given: Q restricted rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
3.67/1.84	   active(__(X, nil)) -> mark(X)
3.67/1.84	   active(__(nil, X)) -> mark(X)
3.67/1.84	   active(U11(tt)) -> mark(U12(tt))
3.67/1.84	   active(U12(tt)) -> mark(tt)
3.67/1.84	   active(isNePal(__(I, __(P, I)))) -> mark(U11(tt))
3.67/1.84	   active(__(X1, X2)) -> __(active(X1), X2)
3.67/1.84	   active(__(X1, X2)) -> __(X1, active(X2))
3.67/1.84	   active(U11(X)) -> U11(active(X))
3.67/1.84	   active(U12(X)) -> U12(active(X))
3.67/1.84	   active(isNePal(X)) -> isNePal(active(X))
3.67/1.84	   __(mark(X1), X2) -> mark(__(X1, X2))
3.67/1.84	   __(X1, mark(X2)) -> mark(__(X1, X2))
3.67/1.84	   U11(mark(X)) -> mark(U11(X))
3.67/1.84	   U12(mark(X)) -> mark(U12(X))
3.67/1.84	   isNePal(mark(X)) -> mark(isNePal(X))
3.67/1.84	   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
3.67/1.84	   proper(nil) -> ok(nil)
3.67/1.84	   proper(U11(X)) -> U11(proper(X))
3.67/1.84	   proper(tt) -> ok(tt)
3.67/1.84	   proper(U12(X)) -> U12(proper(X))
3.67/1.84	   proper(isNePal(X)) -> isNePal(proper(X))
3.67/1.84	   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
3.67/1.84	   U11(ok(X)) -> ok(U11(X))
3.67/1.84	   U12(ok(X)) -> ok(U12(X))
3.67/1.84	   isNePal(ok(X)) -> ok(isNePal(X))
3.67/1.84	   top(mark(X)) -> top(proper(X))
3.67/1.84	   top(ok(X)) -> top(active(X))
3.67/1.84	
3.67/1.84	The set Q consists of the following terms:
3.67/1.84	
3.67/1.84	   active(__(x0, x1))
3.67/1.84	   active(U11(x0))
3.67/1.84	   active(U12(x0))
3.67/1.84	   active(isNePal(x0))
3.67/1.84	   __(mark(x0), x1)
3.67/1.84	   __(x0, mark(x1))
3.67/1.84	   U11(mark(x0))
3.67/1.84	   U12(mark(x0))
3.67/1.84	   isNePal(mark(x0))
3.67/1.84	   proper(__(x0, x1))
3.67/1.84	   proper(nil)
3.67/1.84	   proper(U11(x0))
3.67/1.84	   proper(tt)
3.67/1.84	   proper(U12(x0))
3.67/1.84	   proper(isNePal(x0))
3.67/1.84	   __(ok(x0), ok(x1))
3.67/1.84	   U11(ok(x0))
3.67/1.84	   U12(ok(x0))
3.67/1.84	   isNePal(ok(x0))
3.67/1.84	   top(mark(x0))
3.67/1.84	   top(ok(x0))
3.67/1.84	
3.67/1.84	Special symbols used for the transformation (see [GM04]):
3.67/1.84	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.67/1.84	The replacement map contains the following entries:
3.67/1.84	
3.67/1.84	__: {1, 2}
3.67/1.84	nil: empty set
3.67/1.84	U11: {1}
3.67/1.84	tt: empty set
3.67/1.84	U12: {1}
3.67/1.84	isNePal: {1}
3.67/1.84	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(2)
3.67/1.84	Obligation:
3.67/1.84	Context-sensitive rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   __(__(X, Y), Z) -> __(X, __(Y, Z))
3.67/1.84	   __(X, nil) -> X
3.67/1.84	   __(nil, X) -> X
3.67/1.84	   U11(tt) -> U12(tt)
3.67/1.84	   U12(tt) -> tt
3.67/1.84	   isNePal(__(I, __(P, I))) -> U11(tt)
3.67/1.84	
3.67/1.84	The replacement map contains the following entries:
3.67/1.84	
3.67/1.84	__: {1, 2}
3.67/1.84	nil: empty set
3.67/1.84	U11: {1}
3.67/1.84	tt: empty set
3.67/1.84	U12: {1}
3.67/1.84	isNePal: {1}
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(3) CSRRRRProof (EQUIVALENT)
3.67/1.84	The following CSR is given: Context-sensitive rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   __(__(X, Y), Z) -> __(X, __(Y, Z))
3.67/1.84	   __(X, nil) -> X
3.67/1.84	   __(nil, X) -> X
3.67/1.84	   U11(tt) -> U12(tt)
3.67/1.84	   U12(tt) -> tt
3.67/1.84	   isNePal(__(I, __(P, I))) -> U11(tt)
3.67/1.84	
3.67/1.84	The replacement map contains the following entries:
3.67/1.84	
3.67/1.84	__: {1, 2}
3.67/1.84	nil: empty set
3.67/1.84	U11: {1}
3.67/1.84	tt: empty set
3.67/1.84	U12: {1}
3.67/1.84	isNePal: {1}
3.67/1.84	Used ordering:
3.67/1.84	Polynomial interpretation [POLO]:
3.67/1.84	
3.67/1.84	   POL(U11(x_1)) = x_1
3.67/1.84	   POL(U12(x_1)) = x_1
3.67/1.84	   POL(__(x_1, x_2)) = x_1 + x_2
3.67/1.84	   POL(isNePal(x_1)) = 1 + x_1
3.67/1.84	   POL(nil) = 1
3.67/1.84	   POL(tt) = 1
3.67/1.84	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.84	
3.67/1.84	   __(X, nil) -> X
3.67/1.84	   __(nil, X) -> X
3.67/1.84	
3.67/1.84	
3.67/1.84	
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(4)
3.67/1.84	Obligation:
3.67/1.84	Context-sensitive rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   __(__(X, Y), Z) -> __(X, __(Y, Z))
3.67/1.84	   U11(tt) -> U12(tt)
3.67/1.84	   U12(tt) -> tt
3.67/1.84	   isNePal(__(I, __(P, I))) -> U11(tt)
3.67/1.84	
3.67/1.84	The replacement map contains the following entries:
3.67/1.84	
3.67/1.84	__: {1, 2}
3.67/1.84	U11: {1}
3.67/1.84	tt: empty set
3.67/1.84	U12: {1}
3.67/1.84	isNePal: {1}
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(5) CSRRRRProof (EQUIVALENT)
3.67/1.84	The following CSR is given: Context-sensitive rewrite system:
3.67/1.84	The TRS R consists of the following rules:
3.67/1.84	
3.67/1.84	   __(__(X, Y), Z) -> __(X, __(Y, Z))
3.67/1.84	   U11(tt) -> U12(tt)
3.67/1.84	   U12(tt) -> tt
3.67/1.84	   isNePal(__(I, __(P, I))) -> U11(tt)
3.67/1.84	
3.67/1.84	The replacement map contains the following entries:
3.67/1.84	
3.67/1.84	__: {1, 2}
3.67/1.84	U11: {1}
3.67/1.84	tt: empty set
3.67/1.84	U12: {1}
3.67/1.84	isNePal: {1}
3.67/1.84	Used ordering:
3.67/1.84	Polynomial interpretation [POLO]:
3.67/1.84	
3.67/1.84	   POL(U11(x_1)) = 2 + x_1
3.67/1.84	   POL(U12(x_1)) = 2*x_1
3.67/1.84	   POL(__(x_1, x_2)) = 1 + 2*x_1 + x_2
3.67/1.84	   POL(isNePal(x_1)) = 2*x_1
3.67/1.84	   POL(tt) = 1
3.67/1.84	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.84	
3.67/1.84	   __(__(X, Y), Z) -> __(X, __(Y, Z))
3.67/1.84	   U11(tt) -> U12(tt)
3.67/1.84	   U12(tt) -> tt
3.67/1.84	   isNePal(__(I, __(P, I))) -> U11(tt)
3.67/1.84	
3.67/1.84	
3.67/1.84	
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(6)
3.67/1.84	Obligation:
3.67/1.84	Context-sensitive rewrite system:
3.67/1.84	R is empty.
3.67/1.84	
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(7) RisEmptyProof (EQUIVALENT)
3.67/1.84	The CSR R is empty. Hence, termination is trivially proven.
3.67/1.84	----------------------------------------
3.67/1.84	
3.67/1.84	(8)
3.67/1.84	YES
3.67/1.87	EOF